The Importance of Long-Range Dependence of VBR Video Trac in. Bong K. Ryu

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1 The Importance of Long-Range Dependence of VBR Video Trac in ATM Trac Engineering: Myths and Realities Bong K. Ryu Center for Telecommunications Research, Columbia University, New York, NY Anwar Elwalid Bell Laboratories, Lucent Technologies, Murray Hill, NJ Abstract There has been a growing concern about the potential impact of long-term correlations (second-order statistic) in variable-bit-rate (VBR) video trac on ATM buer dimensioning. Previous studies have shown that video trac exhibits long-range dependence (LRD) (Hurst parameter large than 0.5). We investigate the practical implications of LRD in the context of realistic ATM trac engineering by studying ATM multiplexers of VBR video sources over a range of desirable cell loss rates and buer sizes (maximum delays). Using results based on large deviations theory, we introduce the notion of Critical Time Scale (CTS). For a given buer size, link capacity, and the marginal distribution of frame size, the CTS of a VBR video source is dened as the number of frame correlations that contribute to the cell loss rate. In other words, second-order behavior at the time scale beyond the CTS does not signicantly aect the network performance. We show that whether the video source model is Markov or has LRD, its CTS is nite, attains a small value for small buer, and is a non-decreasing function of buer size. Numerical results show that (i) even in the presence of LRD, long-term correlations do not have signicant impact on the cell loss rate; and (ii) short-term correlations have dominant eect on cell loss rate, and therefore, welldesigned Markov trac models are eective for predicting Quality of Service (QOS) of LRD VBR video trac. Therefore, we conclude that it is unnecessary to capture the longterm correlations of a real-time VBR video source under realistic ATM buer dimensioning scenarios as far as the cell loss rates and maximum buer delays are concerned. 1 Introduction There has been a growing concern about the potential impact of long-term correlations of variable-bit-rate (VBR) video trac on ATM trac engineering. This concern is based on two recent ndings: (i) VBR video trac exhibits long-range dependence (Hurst parameter larger than 0.5) [2]; (ii) the buer overow probability (BOP) 1 of long-range dependent (LRD) video trac decays hyperbolically [14, 18] or in a Weibull fashion [5, 17]. A direct implication of (ii) is that the notion of eective bandwidth based on Markov models of which the BOP decays exponentially does not apply to LRD trac (see Section 4 for detail). This has led many researchers and practitioners to believe that the LRD property of VBR video trac will have critical impact on ATM trac engineering in general. Dierent, and perhaps opposing viewpoints about the impact of the LRD property also exist, however. Heyman and Lakshman [8] and Elwalid et al [6] analyzed several traces of VBR videoconferencing trac and modeled them using the rst-order Discrete Autoregressive processes [] (see Section 3.1 for detail) by matching the marginal distribution of frame size and rst-lag correlation. Despite the fact that these traces were shown to possess longrange dependence by Beran et al [2], Elwalid et al showed that for practical ranges of buer size and cell loss rate (CLR) 1, the model, which is a short-range dependent (SRD) process, provides an accurate prediction of CLRs for the purpose of real-time connection acceptance control. Later, we fully explain this result, which appears to contradict the relevant results that appear in the literature (see Section 4), by considering the realities that (i) the autocorrelations of these traces drop quickly for small lags; and (ii) the buer size required for multiplexing many VBR video sources is typically small due to the delay constraint imposed on real-time applications. At least two claims have been made in the literature regarding the importance of the LRD property inatm trac 1 Throughout this paper, we use buer overow probability (BOP) to represent the probability that buer content exceeds some amount in an innite buer system, and cell loss rate (CLR) to represent the fraction of lost cells in a nite buer system.

2 engineering: 1. While long-term correlations of LRD processes are individually small, their cumulative eect on the CLR is non-negligible [2]. 2. The buer behavior of LRD VBR video trac cannot be accurately predicted by simple, parsimonious Markov-based (or SRD in general) models [2, 10, 23]. The focus of this study is two-fold: in a narrow sense, we check the validity of the above two claims under realistic scenarios of ATM buer dimensioning; and in a broad sense, we explore a general relationship between buer size and video frame correlations (with other system parameters held xed). We consider ATM multiplexers of real-time VBR video sources over the following ranges: buer size (maximum delay) less than 20 { 30 msec, and CLR less than In other words, we choose a link capacity (or utilization) such that the system operating point lies within the above ranges of buer size and CLR. We envision that the total end-to-end delay (propagation + processing + network queueing + others) allowed for typical real-time VBR video applications is about 200 msec or so, or even smaller. Since such applications are most likely to pass along several network nodes, we believe that the delay at each node should be kept within 20 { 30 msec at maximum. Using results from large deviations theory, we show that: (i) long-term correlations do not have signicant impact on CLR; and (ii) (ii) short-term correlations have dominant eect on cell loss rate, and therefore, well-designed Markov trac models are eective for predicting Quality of Service (QOS) of LRD VBR video trac. These results, veried by simulation, disprove the above claims under practical ranges of buer size and CLR. Since the key issue of this study is to investigate the eect of dierent types of autocorrelation structures (short-range dependence or long-range dependence) on the multiplexer performance, it is essential that we have models of which the frame correlations can be easily and exibly adjusted. For this reason, we use stochastic VBR video models, unlike the previous studies in which real video traces were used. In particular, we construct LRD video models such that they resemble real traces exhibiting LRD; see [8] for examples of such traces. Further, in order to exclude the eect of the marginal distribution (of frame size) on buer behavior, we set the marginal distributions of all models to be identical so that the rst-order statistics do not contribute to the difference in queueing behavior. Wechoose Gaussian frame size distribution to simplify our analysis. It is noted that the Gaussian frame size distribution may not be a universal characterization of video traces; some traces have been found to possess heavier tails than that of the Gaussian distribution [7]. As discussed in Section 6, however, we believe that our main results are unlikely to be aected by this assumption. Based on the results from large deviations theory, we introduce the notion of Critical Time Scale (CTS) which describes a relationship between buer size and frame correlations (with other system parameters such as link capacity and marginal distribution of frame size held xed). For given buer size, the CTS of a VBR video source is dened as the number of frame correlations that contribute to the CLR. For example, for VBR video sources with Gaussian marginal distribution of frame size, knowledge of CTS implies exactly how many frame correlations aect the cell loss rate (CLR); correlations after such lag (CTS) do not contribute to CLR. Later we show that whether the model is Markov or has the LRD property, its CTS is nite, attains a small value for small buer, and is a non-decreasing function of the buer size. In other words, under realistic scenarios of ATM buer dimensioning, the number of frame correlations which aect BOP is nite and small even in the presence of the LRD property. As a result, it is unnecessary to capture the long-term correlations of a real-time VBR video source as far as the cell loss rates and maximum buer delays are concerned as QOS metrics. The rest of this paper is organized as follows. Section 2 provides two denitions of an LRD process used in this paper. Section 3 presents the details of two stochastic processes: Discrete-AutoRegressive process of order p [DAR(p)] [9] and Fractal-Binomial-Noise-Driven Poisson point process (FBNDP) [19]. Four VBR trac source models, V v, Z a, S, and L, are constructed using the above processes and their marginal distributions and autocorrelations are discussed. Section 4 provides an overview of results from large deviations theory on the BOP of ATM multiplexers. Using these analytical tools, we examine the two claims by comparing the CTSs and BOPs of the four video models. Section 5.5 provides simulation results which verify the analytical results of Section 4. Finally, conclusions and discussions are presented in Section 6. 2 Long-Range Dependence: Denitions Let X = fx k : k 1g be a wide-sense stationary (WSS) process in the discrete-time domain with E[X k], variance 2 E[(X k ) 2 ], and autocorrelation function r(k; T s) E[(X n )(X n+k )]= 2 : X k represents the size of the k-th frame in cells with frame duration of T s sec. We call X an asymptotic LRD process if the tail of its autocorrelation function r(k; T s) is given by 2 [10]: r(k; T s) k (2 2H) ; as k!1. (1) The quantity H is called the Hurst parameter and completely characterizes the relation (1). Fractional autoregressive integrated moving average models, F-ARIMA(p,d,q), are an example of asymptotic LRD processes [10]. Likewise, we call X an exact LRD process if r(k; T s)= 1 2 g(ts)r2 (k 2H ) for k>0 (2) 2 The symbol denotes an asymptotic relation.

3 with 0:5 < H <1, 0 g() 1 independent of k, and r 2 (h(k)) h(k +1) 2h(k)+h(k 1), the second central dierence operator. Due to the asymptotic equivalence of dierencing and dierentiation, (2) can be approximated to r(k; T s) g(t s)h(2h 1)k (2 2H) (3) for large k. The discrete-time fractional Gaussian Noise (FGN) process has an autocorrelation function of the form (2) with g(t s) = 1 [21], and therefore is an exact LRD process. A family of fractal stochastic point processes (FSPPs) introduced in [19, 20] also yield exact LRD processes in the sense of (2) with g(t s)=ts 2H 1 =(Ts 2H 1 +T 2H 1 0 ). T 0 is called fractal onset time which is used to control the variance of the frame size; see [20] for detail. We distinguish (1) and (2) because we feel that the rst claim seems to stem from (1) which completely ignores the short-term correlations of X. 3 LRD and SRD VBR Video Models We check the validity of the two claims given in the introduction by studying two relevant issues, one for each claim, respectively: I: Eect of short- and long-term (frame) correlations on CLRs. II: Ecacy of Markov models in predicting the loss characteristic of LRD VBR video trac. The objective of studying the rst issue is to examine whether the long-term correlations have signicant impact on CLRs in the presence of the LRD property. For this purpose, we use two asymptotic LRD models: V v and Z a. The long-term correlations of V v are controlled by v, while its short-term correlations (at least the rst-lag correlation) are unchanged. On the other hand, the short-term correlations of Z a are controlled by a, while its long-term correlations (Hurst parameter) are unchanged; see Fig. 1. We use the Autocorrelation r(k) controlled by a (Z a ) short-term Lag (k) controlled by v (V v ) long-term Figure 1: Eect of the change of a and v on the autocorrelation function of the models Z a and V v. superposition of the Fractal-Binomial-Noise-Driven Poisson point process (FBNDP) [20] and process [9] for constructing V v and Z a. Details on how we construct these two models are given in Section 3.3. For the rst claim to be valid, we must observe, under practical scenarios of ATM buer dimensioning, that: (i) the CLRs of V v for dierent values of v are signicantly dierent; and (ii) the CLRs of Z a for dierent values of a are close to each other. Later, we show that what we observe is the opposite, indicating that even in the presence of the LRD property, it is the shortterm correlations that have a dominant impact on CLRs under realistic scenarios of ATM buer dimensioning, not the long-term correlations. The objective of studying the second issue is to investigate whether a simple, well designed Markov model can provide a good prediction of the CLRs of an LRD VBR video source. For this purpose, we use two additional models: a short-range dependent (SRD) model S, and an exact LRD model L. For a given asymptotic LRD model Z a,we construct S such that the rst p correlations of Z a are exactly matched. Also, we use an exact LRD model L to match only the long-term correlations of Z a. We use DAR(p) process for S and the FBNDP process for L. In summary, Z a serves the role of a real trace that exhibits the LRD property; S [DAR(p)] captures only the short-term correlations of Z a ; and L only the long-term correlations of Z a ; see [9] and [20, chapter 6] on how DAR(p) processes capture the rst p correlations of Z a. For the second claim to be valid, we must observe that L outperforms S in predicting the CLR of Z a over practical ranges of buer sizes and CLRs. Later, we show that when a system operating point is chosen within practical ranges of buer sizes and CLRs, DAR(p) models perform superior to L even for small p (p =1;2;3). The crucial feature of these four models (V v, Z a, S, and L) is that their marginal distributions of frame size are identical, implying that the rst-order statistics do not contribute to the dierence in buer behavior. As discussed in the introduction, we choose the Gaussian distribution since the marginal distribution is not the concern of this study. 3.1 DAR(p) The DAR(p) process (discrete autoregressive process of order p), constructed and analyzed by Jacobs and Lewis [9], is a p-th order Markov chain with the physical and correlation structure of a p-th order autoregressive process; S n depends explicitly on S n 1;:::;S n p. The process is specied by the stationary marginal distribution of fs ng and several other chosen parameters which, independently of the marginal distribution, determine the correlation structure. This DAR(p) process is dened as follows. Let f ng be a sequence of i.i.d. random variables taking values in Z, the set of integers, with distribution. Let fv ng be a sequence of Bernoulli random variables with P(V n =1)=1 P(V n = 0) = for 0 <1. For the process, represents the rst-lag autocorrelation. Let fa ng be a sequence of i.i.d. random variables taking values in P f1; 2;:::;pg with P(A n = p i) =a i0; i=1;2;:::;p with ai = 1. Let i=1 S n = V ns n An +(1 V n) n (4)

4 for n =1;2;:::. Then, the process S = fs ng is called the DAR(p) process. Note that this process has p degrees of freedom, and therefore has the capability to match upto the rst p autocorrelations. a 3.2 Fractal-Binomial-Noise-Driven Poisson Process (FB- NDP) The FBNDP model is described in [20] in detail. Hence, only a brief description and relevant statistics of the model is provided here. First, we obtain a fractal ON/OFF process with ON/OFF periods independently and identically distributed by the same heavy-tailed density function as p(t) = A 1 e t=a for t A, e A t (+1) for t>a, with = 2 (1 <<2). Then, M independent and identical fractal ON/OFF processes are added, yielding a fractal binomial process (FBN). This serves as a stationary stochastic rate function for a Poisson process; the FBNDP results [20]. This model has four parameters:, A, M, and R (the rate of a fractal ON/OFF process when it is ON) [20]. They determine the relevant statistics of the FBNDP as H = ( +1)=2 = RM=2 T 0 = ( + 1)(2 ) 1 (1 )e 2 +1 R 1 A 1 1= where H is the Hurst parameter, the mean arrival rate (cells/sec), and T 0 the fractal onset time (sec). Denote by N(t) the counting process (i.e., number of arrivals up to time t) constructed from the FBNDP. If an exact LRD process L = fl ng is constructed from the FBNDP as then we have L n N[nT s] N[(n 1)T s]; L E[L n] = T s L 2 Var[L n] = [1 + (T s=t 0) ] T s r L(k) Cov(L n;l n+k)=l 2 = Ts Ts +T r2 (k +1 ) where T s is the frame duration and r 2 the second central dierence operator. Since M is an extra parameter, it is used to control the marginal distribution of fl ng. In fact, for large M and L, the marginal distribution of fl ng approaches a Gaussian form with mean L and variance L 2 by the central limit theorem [15]. 3.3 fv v, Z a g = FBNDP + Many analyses of VBR real traces show that their short-term correlations are characterized by geometric decay [8]. When those traces exhibit the LRD property, mathematically it is represented as power-law decaying long-term correlations; see (1). In order to make V v and Z a represent a wide range of real LRD VBR traces in terms of frame correlations, we use the superposition of the FBNDP and processes for constructing Z a and V v for which the process contributes to geometric decay for small lags and the FB- NDP to power-law decay for large lags; see Figs. 3 (a) and (b). Let the process X = fx ng be constructed from the FB- NDP with a Gaussian marginal distribution with ( X, 2 X) and Y = fy ng from also with a Gaussian marginal distribution with ( Y, 2 Y ), and assume X and Y are independent. Then, both the Z a and V v also have a Gaussian marginal distribution with = X + Y 2 = X 2 + Y 2 r(k) = v v +1 rx(k)+ 1 v+1 ry(k) = v v+1 T s (T s +T 0 ) 1 2 r2 (k +1 )+ 1 v+1 ak ;(5) with a the rst-lag correlation of the component and v 2 X= 2 Y : Note that r(k) is expressed as a weighted sum of r X(k) and r Y (k). The fact that there exists k 0 < 1 such that r X(k) >r Y(k) for k>k 0, regardless of the choice of, a, and v (> 0), makes Z a and V v asymptotic LRD processes; see Fig. 1. Indeed, a and v in (5) are the same as in Z a and V v, respectively. For V v,wexh(or ) and change a such that for dierent values of v, the rst-lag correlation is identical and the next several correlations are very close to each other; see Table 1 and Fig. 3-(a) for the resulting parameters and autocorrelations. For Z a,we again x H and change a to obtain diverse behavior of short-term correlations. In this case, we set v = 1 (and thus 2 X = 2 Y ) and X = Y so that both the FBNDP and processes contribute equally to the mean and variance of Z a. 4 Large Deviations Theory and Critical Time Scale 4.1 Brief history on large buer asymptotics for LRD processes The rst result on queueing analysis of self-similar trac seems to appear in Norros [17] in which the popular Weibull (lower) bound on the BOP has been established using fractional Brownian Motion. Also, using the techniques of large deviations theory, Dueld et al [5] have shown that under mild conditions, the BOP of LRD trac, whether exact or asymptotic, is again approximated by Weibull behavior as the buer size goes to innity. While the above results exhibit Weibull decays, Likhanov et al [14] and Parulekar and Makowski [18] show that for the M=G=1-type model of Cox [4], the tail asymptotic of the BOP decays at most hyperbolically. We provide another Weibull approximation for statistically identical N Gaussian exact LRD processes, which isgiven by h i 1 P(W >B)exp J(N; b; c) 2 log f4j(n; b; c)g (6)

5 with J(N; b; c) N 2H 1 (c ) 2H 2g(T s) 2 (H) 2 B2 2H where (H) H H (1 H) 1 H, g(t s) is from (2), N is the number of homogeneous sources multiplexed, is the mean frame size per source (cells/frame), 2 is the variance of frame size, and c is the bandwidth per source (cells/frame). The proof is provided in the Appendix. 3 Note that for H =1=2 and large N, (6) is reduced to a familiar log-linear behavior as predicted by the well-known technique of eective bandwidth based on Markov models. Indeed, one may think of (6) as the result of the rst claim and the cause of the second claim. Also, it serves as a basis for the argument that the popular eective bandwidth scheme based on simple Markovian trac models would consequently provide a poor estimate of CLRs of LRD trac. 4.2 Large buer asymptotics and the Critical Time Scale for Gaussian sources Large deviations theory has served as a powerful tool for analyzing the performance of communications networks [22, references therein]. We use the Bahadur-Rao (B-R) Asymptotic of buer overow probability of an ATM multiplexer with N sources [16]. We use the following notations [3, 16]. Denote by C the constant service rate during the frame duration measured in cells/frame and by fx ng the input process (i.e., the size of n-th frame from N identical sources) in cells. The workload at the start of n-th frame interval is denoted by W n and W n+1 = (minf(w n + X n C);Bg) + where B is a buer size (cells) and x + max(x; 0). Let b and c denote respectively the amounts of buer space and bandwidth per source, so that B = Nb and C = Nc. Suppose fx ng is the superposition of N identical sources with Gaussian marginal, each distributed as fy ng with mean, variance 2, and autocorrelation function r(k). Then, the BOP (c; b; N) predicted by the B-R asymptotic takes the following form [16] with and (c; b; N) exp ( NI(c; b)+g 1(c; b; N)) (7) I(c; b) [b + m(c )] 2 inf ; m1 2V (m) (8) g 1(c; b; N) 1=2 log[4ni(c; b)] (9) V (m) Var mx i=1 = 2 "m +2 Y i! mx i=1 (m i)r(i) : (10) 3 We note that similar expression (without the second term in the exponent) has been obtained by Parulekar and Makowski [18] for the discrete-time fractional Gaussian Noise (FGN) process (and thus g(t s) = 1) using the results of Dueld et al [5]. Note that ignoring g 1(c; b; N) yields the Large N Asymptotic given by Courcoubetis and Weber [3]. Let's examine (8) and (10) with a closer look. Let f(c; b; m) [b + m(c )] 2 and denote by m b the value of m at which the inmum of f(c; b; m)=2v (m) is obtained, i.e., m b arginf m1 f(c; b; m)=2v (m). Assuming that m b is unique [16], for xed b, c, and, m b represents the number of frame correlations that completely determine (c; b; N). More importantly, it implies that the rst m b frame correlations are only meaningful in evaluating (c; b; N) through V (m b). For this reason, we call m b Critical Time Scale (CTS). It follows that the LRD property described by (1) and (2) must force m b to be (at least) very large even for small b, in order to support the rst claim. We rst show that m b is nite in most practical cases. Assuming that the autocorrelation function r(k) is monotonically decreasing, we note that both f(c; b; m) and V (m) are increasing functions of m. Also, the increasing rate of V (m) is slower than m 2, i.e., V (m) 2 [m + O(m 2 )] for SRD processes [8] 2 g(t s)m 2H for exact LRD processes for large m [4, 8]. It follows that for xed b, lim f(c; b; m)=2v (m) =1; m!1 yielding m b < 1. This result immediately indicates that for nite buer size, the cumulative eect of autocorrelations is bounded regardless of the presence of the LRD property, disproving the rst claim. Second, our investigation further indicates that m b is actually an order of b, i.e., m b =b K for some constant K. For example, for a relatively large m b, it has been found that K =1=(c ) for a Gaussian AR(1) process [3] and K = H (1 H)(c ) for a Gaussian exact LRD process (see Appendix). 4 Moreover, m 0 is always equal to 1, indicating that correlations do not aect the CLR at all when buer size is zero. [3, 6]. From this, we stipulate that for small b, m b would be also small. The following section provides numerical examples which show that whether the model is SRD (Markov) or LRD, its CTS (m b) is nite, attains a small value for small buer, and is a non-decreasing function of buer size b. These results are converged to that (i) the Weibull behavior in (6) is not signicant for realistic buer size; and (ii) DAR(p) processes capturing short-term correlations of Z a provide a satisfactory approximation of CLRs of Z a. 5 Numerical Results 5.1 Parameter specication We choose the model parameters of V v, Z a, S, and L as follows. 4 We note that the same result has been independently obtained by Addie et al [1] using the superposition of two Gaussian AR(1) processes.

6 1. The frame rate is chosen to be 25 frames/sec, yielding frame duration T s =0:04 (sec). 2. The marginal distribution of all the models is Gaussian with mean () 500 (cells/frame) and variance ( 2 ) 5000 (cells/frame) 2. The choice of M = 15 for Z a and V v and M =30for Lfor the underlying process FBNDP, respectively, are found to provide good approximations of the Gaussian marginal distribution for simulation purpose. 3. For V v, three values of v are used: 0.67, 1, and 1.5. Corresponding values of a are determined such that all the V 0:67, V 1, and V 1:5 have the same rst-lag correlation; see Fig. 3-(a). Frame Size (cells) For Z a, four values of a are used: 0:7; 0:9; 0:975; 0:99. Note that once,, T 0, and M are xed, the marginal distribution of Z a is not aected by a. 5. For each a of Z a,we determine the parameters of the DAR(p) model (S) such that it exactly matches the rst p correlations of Z a using the steps given in [20, chapter 6]. 6. The value of for the FBNDP component ofz a was chosen as 0:8, yielding the Hurst parameter H =0:9. 7. The value of for L was obtained such that the tail of r L(k; T s) provides the best t to that of r Z a(k; T s). Note that because of the v=(v + 1) factor in (5), one cannot use the same of Z a for L in order to make their long-term correlations identical. The choice of =0:72 (H =0:86) was found to be suciently accurate for the purpose of this study; see Fig. 3-(b). 8. Finally, the values of T 0 for Z a and L are determined from the given mean, variance, and of each model. Table 1 summarizes the values of all the parameters of each model. We note that the dierent choice of key parameters such ashyields the qualitatively same result as in Table 1. Fig. 2 shows sample paths of Z 0:7 and its corresponding model which matches the rst-lag correlation of Z 0:7. Strong low-frequency behavior (thus long-term correlations) is clearly visible in Z 0:7. In particular, the burst-within-burst structure of self-similar trac, i.e., traf- c spikes riding on longer-term ripples, that in turn ride on still longer term swells [10], is easily observed. On the other hand, such characteristic is not observed in the model, as it is a short-range dependent model (Hurst parameter is 0.5). Instead, the process captures the fast-time scale behavior of Z 0:7 quite closely. 5.2 Autocorrelations With the parameter specication given in the previous section, we examine the autocorrelations of V v, Z a and L for a wide range of time scales. Fig. 3-(a) shows that the shortterm correlations of V v are very close to each other (up to Frame Number Figure 2: Sample path comparison of Z 0:7 and matched for 10 sources multiplexed (N = 10). about ve lags), and in particular, their rst-lag correlation is identical. Fig. 3-(b) shows that (i) the larger value of a of Z a yields stronger short-term correlations; and (ii) the longterm correlations of both Z a and L are very close up to at least 1; 000 lags. We note that several traces of VBR videoconferencing trac exhibit the correlation structure similar to Z a (or V v )[6, 8]. Figs. 3-(c) and (d) show that indeed the DAR(p) processes match the rst p correlations of Z a. 5.3 Critical time scales Let us examine the eect of dierent correlation structures (SRD or LRD) on the CTS (m b). Fig. 4-(a) compares m b of V v for three dierent values of v. Recall that the short-term correlations of V v are very close to each other; see Fig. 3- (a). As a result, the values of their m b are much the same for small buer. On the other hand, m b of Z a for dierent values of a shows signicant dierence, as many as 15 even at B = 2 (msec). These results clearly indicate that it is the short-term correlations which have dominant impact on the CTS, not the long-term correlations. Both gures also illustrate that m b attains a small value for small b (or B) and non-decreasing function of b (or B). Moreover, we observe that the process with stronger short-term correlations (higher a) yield larger m b for the same size of buer in spite of the identical long-term correlations. 5.4 Large buer asymptotics Figs. 5-(a) and (b) compare the buer overow probabilities (BOPs) of V v and Z a for N = 30 based on the B-R asymptotic (7). As predicted by Fig. 4, these gures clearly show that short-term correlations are dominant in determining BOPs (or CLRs) for small buer. Close behavior in short-term correlations yield close loss probabilities [Fig. 5- (a)], while drastically dierent short-term correlations yield

7 v a (cells/sec) T 0 (msec) M V v Z a , 0.9, 0.975, L a 0:7 0:975 S =0:82, a 1 =1 =0:68, a 1 =1 =0:87, a 1 =0:70, a 2 =0:3 =0:72, a 1 =0:84, a 2 =0:16 =0:89, a 1 =0:63, a 2 =0:18, a 3 =0:19 =0:73, a 1 =0:82, a 2 =0:10, a 3 =0:08 Table 1: Specication of model parameters of V v, Z a, S, and L. For S, two cases of Z a are considered: Z 0:7 and Z 0:975. Autocorrelation r(k) v = 0.67 v = 1.0 v = 1.5 Autocorrelation L with H 0.86 with 0.7 with 0.9 with Z with a = Lag k (time unit Ts = 40 msec) Lag k (time unit Ts = 40 msec) (a) Correlations of V v for dierent values of v (b) Correlations of Z a for dierent values of a Autocorrelation Autocorrelation Z with a = DAR(4) DAR(25) Lag k (time unit Ts = 40 msec) (c) Z 0:7 and corresponding S (DAR(p)) Lag k (time unit Ts = 40 msec) (d) Z 0:975 and corresponding S (DAR(p)) Figure 3: Analytic autocorrelation functions of V v, Z a, S, and L

8 signicant discrepancy despite of having identical long-term correlations [Fig. 5-(b)]. Also, stronger correlations of an arrival process yield slower decaying BOP for the range of buer size of interest. This illustrates why the rst claim is not valid under practical ranges of buer size and CLR. We note that eventually the curves in Fig. 5-(b) will exhibit the same decaying characteristic when buer size becomes larger and larger, for they all have the same Hurst parameter [5]; however, such range of the buer size is beyond the practical consideration; see Fig. 7. minimizer m*(b) v = 0.67 v = 1.0 v = Total (a) V v models are indeed capable of providing good approximations of BOP (or CLRs) of an LRD process for the practical ranges of buer size and CLR. Fig. 6-(b) compares the B-R asymptotics of Z 0:7 and DAR(p), p = 1;2;3. Recall that all of their autocorrelation functions drop very quickly at small lags (see Figs. 3-(b) and (c)), yielding close loss behavior as we have seen from V v. This type of autocorrelations has been observed in several VBR videoconferencing traces studied in [6, 8]. We further observe that the dierence between all the curves at the loss rate 10 6 is only within the order of one. This dierence becomes negligible when the loss rate is translated to the number of admissible VBR video connections, which iswhy the model provides accurate prediction of the number of admissible connections for LRD traces [6]. Log10[ Buffer Overflow Probability ] v = 0.67 v = 1.0 v = 1.5 minimizer m*(b) with 0.7 with 0.9 with Z with a = Total (b) Z a Figure 4: Minimizer m b vs total buer size B. (a) eect of same short-term correlations on m b, (b) eect of same longterm correlations on m b. In both cases, c and are xed as c = 526 and = 500, and N = 100. Fig. 6-(a) compares the BOPs of Z 0:975,DAR(p), p = 1; 2; 3, and L again based on the B-R asymptotic (7). It is easy to observe that even the model outperforms L for a wide range of buer size of interest. Moreover, notice that the curve of each DAR(p) model is getting closer to that of Z 0:975 as p increases, implying that capturing more and more short-term correlations yields more accurate prediction of CLR. This clearly breaks the second claim; simple Markov Log10[ Buffer Overflow Probability ] (a) V v Z with a = 0.9 Z with a = Z with a = (b) Z a Figure 5: Comparison of BOPs of V v and Z a based on the Bahadur-Rao asymptotic with N = 30 and c = 538 (cells/frame): eect of short- and long-term correlations on the cell loss. We provide one more evidence illustrating the impact of correlation structure on loss behavior. Figs. 7-(a) and (b) repeat Figs. 6-(a) and (b) over much wider range of

9 buer size, even for unrealistically large buer sizes. The box in the upper left corner of each gure corresponds to the ranges of buer size and CLR over which Figs. 6-(a) and (b) are drawn. Indeed, these two gures show where the two claims are coming from; L eventually outperforms Markov models over the range of buer size which is beyond practical consideration. Observe that the decaying rates of Z a follow that of L from about B = 40 msec, which serves as another pictorial proof that Z a and L have the same long-term correlations. All of these arguments converge to the point that future trac analysis should focus more on nding appropriate time scale at which trac behavior is to be captured, rather than on providing accurate trac models. The following section provides simulation results, verifying the results of this section. Log10[ Buffer Overflow Probability ] Log10[ Buffer Overflow Probability ] Z with a = L with H = (a) Z 0:975,DAR(p), L (b) Z 0:7,DAR(p) Figure 6: Comparison of BOPs of Z a,dar(p), and L with N = 30 and c = 538: ecacy of simple Markov models. 5.5 Simulation For each of the four models we run 60 replications, each of which generates half a million frames. This extensive amount of computation ensures accurate and numerically condent estimations which may not be otherwise obtained due to the heavy-tailed ON/OFF times of the FBNDP model. We assume that the beginning of frame of each source is same and that cells are equispaced over the frame duration (deterministic smoothing). For a given buer size, we estimate the fraction of lost cells (nite buer system). Log10[ Buffer Overflow Probability ] Log10[ Buffer Overflow Probability ] Z with a = L with H = (a) Z 0:975,DAR(p), L (b) Z 0:7,DAR(p) Figure 7: Comparison of BOPs of Z a, L, and corresponding DAR(p) over a wide range of buer size with N = 30 and c = 538. Most importantly, both Figs. 8 and 9 conrm the results predicted by Figs. 5 and 6, respectively. We note that all the CLR curves begin around the same value at zero buer (slightly larger than 10 5 ) which conrms that the parameters given in Table 1 produce the same Gaussian marginal distribution of frame size for all four models. Other results with dierent choices of N and c, which are omitted here, show qualitatively the same results as Figs. 8 and 9, further supporting our conclusions. Fig. 10 compares the two large buer asymptotics (B-R and large N asymptotics) with simulation results for model matched to Z 0:975. First, we observe that all the

10 curves are parallel under the ranges of buer size and CLR of interest, indicating that both asymptotics capture the loss behavior of a given model in a qualitative sense. Second, the B-R asymptotic provides tighter upper bound for the CLR than the large N asymptotic, with about one order of magnitude improvement in this case. However, the dierence between the B-R asymptotic and the estimated CLR is still about 2 orders of magnitude, posing an open question of the applicability of large buer asymptotics (innite buer) for approximating CLRs (nite buer). Log10[ Cell Loss Rate (CLR) ] Log10[ Cell Loss Rate (CLR) ] V with v = 0.67 V with v = 1.0 V with v = (a) V v Z with a = (b) Z a Figure 8: Cell loss rates (CLRs) of V v and Z a from simulation (nite buer). N = 30 and c= 538 (cells/frame): eect of short- and long-term correlations on the cell loss (corresponding to Fig. 5). 6 Concluding Remarks This work has looked at the impact of the autocorrelations on the loss behavior of real-time VBR video trac with particular emphasis on the practical implications of LRD. There exists a growing concern about the potential impact of LRD on the network performance due to the following reasons: (i) previous studies on network performance analysis such as admission control have based on the assumption of Markov trac models almost exclusively, whereas real video traces exhibit LRD; (ii) traditional Markovian models lack the capability of capturing this property in an economical way. This study reveals that such property is not practically important in determining cell loss rates (CLRs) under the realistic scenarios of ATM buer dimensioning; shortterm correlations (high-frequency behavior) have dominant impact on network performance. By analyzing and comparing the critical time scales (CTSs) of carefully designed VBR video models, we have been able to show that under realistic ranges of buer size and CLR, the number of frame correlations which aect the cell loss is nite and small even in the presence of LRD. Simulation results verify that the two claims on the potential impact of the LRD property are not valid (myths) under the practical ranges of buer size and CLR. Log10[ Cell Loss Probability ] Log10[ Cell Loss Probability ] Z with a = L with H = (a) Z 0: (b) Z 0:7 Figure 9: Comparison of the CLRs of Z a, L, and matched DAR(p) from simulation. N = 30, c = 538 (cells/frame): ecacy of simple Markov models (corresponding to Fig. 6).

11 Log10[ BOP ] or Log10[ CLR ] large N asymptotic Bahadur-Rao asymptotic Simulation (infinite buffer) Simulation (finite buffer) Figure 10: Accuracy of two large buer asymptotics. The video model is matching Z 0:975. N = 30 and c = 538. We conclude this work with brief discussions on the following relevant issues. 6.1 Eect of other marginal distributions Strictly speaking, our results are based on the Gaussian marginal distribution, which is considered as having the \lightest" tail. Unfortunately, other distributions such as heavytailed frame sizes pose diculty in mathematical analysis of queue performance. For the negative binomial marginal distribution, the same conclusion was obtained by Heyman and Lakshman using a slightly dierent approach [8]. In general, the required bandwidth for distributions with heavier tails will be larger than for Gaussian distribution to keep the operating point unchanged (within the practical ranges of buer size and CLR). Once the bandwidth is properly chosen, and all the video models have the corresponding same marginal distribution of frame size, then the dierence in buer behavior will be again caused by the dierence in their higher-order statistics, mainly the autocorrelations [13]. Therefore, we expect that our conclusions are unlikely to be signicantly aected by other marginal distributions, though future work might be needed to support this. 6.2 Multiple-time-scale based trac analysis There is a signicant amount of interest in capturing the time scale at which key statistics of trac to network performance are to be evaluated [11, 12, 13, 16]. Our conclusions clearly indicate that trac behavior after certain time scale (i.e., CTS) is not relevant to network performance such as CLR. As discussed in [16], the CTS is closely related with the cuto frequency! c introduced in [11, 12, 13]. Note that a practical buer size is about one frame duration (30 msec or so), whereas the time scale at which the LRD property of VBR video trac begins to appear is an order of tens, hundreds, or even thousands frames. Further work is currently under way on nding CTS of various types of trac sources including MPEG-coded video, and on applying it for trac management in broadband networks. Acknowledgement The authors would like to thank anonymous reviewers for their constructive comments which signicantly improved the presentation of this paper. They are also very grateful to Debasis Mitra for helpful discussions and insights he provided during the course of this work. This work was completed while the rst author was on leave with Bell Laboratories, Murray Hill, NJ. Appendix: B-R Asymptotic for N Gaussian LRD sources In this appendix, we derive (6) using the Bahadur-Rao (B- R) asymptotic [16]. Recall that for an exact LRD process L = fl 1;L 2;:::g with mean and variance 2, its autocorrelation function (ACF) is given by r(k; T s) = 1 2 g(ts)r2 (k 2H ) for k>0 H(2H 1)g(T s)k 2H 2 due to the equivalence of dierencing and dierentiation. Then for large m, we approximate V (m) dened by (10) as mx V (m) "m 2 +2H(2H 1)g(T s) (m k)k 2H 2 k=1 Z m m 2 +2H(2H 1)g(T s) (m x)x 2H 2 2 g(t s)m 2H : (11) The above approximation has been found to be accurate even for small m [4]. Substituting V (m) in the rate function (8) with (11) yields, For xed b and c, let [b + m(c )] 2 I(c; b) inf m1 2 2 g(t s)m 2H : [b +(c h(x) )x]2 2 g(t s)x 2H : Dierentiating h(x) and setting the result to 0 gives its root x at x H = (1 H)(c ) b m b: Then, I(c; b) is given by I(c; b) (c ) 2H 2g(T s) 2 (H) 2 b2 2H : Employing B = Nb and substituting the result to (7) yields (6). 1

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